A Process-based Model with Temperature, Water, and Lab-derived Data Improves Predictions of Daily Culex pipiens/restuans Mosquito Density

While the number of human cases of mosquito-borne diseases has increased in North America in the last decade, accurate modeling of mosquito population density has remained a challenge. Longitudinal mosquito trap data over the many years needed for model calibration, and validation is relatively rare. In particular, capturing the relative changes in mosquito abundance across seasons is necessary for predicting the risk of disease spread as it varies from year to year. We developed a discrete, semi-stochastic, mechanistic process-based mosquito population model that captures life-cycle egg, larva, pupa, adult stages, and diapause for Culex pipiens (Diptera, Culicidae) and Culex restuans (Diptera, Culicidae) mosquito populations. This model combines known models for development and survival into a fully connected age-structured model that can reproduce mosquito population dynamics. Mosquito development through these stages is a function of time, temperature, daylight hours, and aquatic habitat availability. The time-dependent parameters are informed by both laboratory studies and mosquito trap data from the Greater Toronto Area. The model incorporates city-wide water-body gauge and precipitation data as a proxy for aquatic habitat. This approach accounts for the nonlinear interaction of temperature and aquatic habitat variability on the mosquito life stages. We demonstrate that the full model predicts the yearly variations in mosquito populations better than a statistical model using the same data sources. This improvement in modeling mosquito abundance can help guide interventions for reducing mosquito abundance in mitigating mosquito-borne diseases like West Nile virus.

Generating a heterosexual bipartite network embedded in social network

We describe an approach to generate a heterosexual network with a prescribed joint-degree distribution embedded in a prescribed large-scale social contact network. The structure of a sexual network plays an important role in how all sexually transmitted infections (STIs) spread. Generating an ensemble of networks that mimics the real-world is crucial to evaluating robust mitigation strategies for controlling STIs. Most of the current algorithms to generate sexual networks only use sexual activity data, such as the number of partners per month, to generate the sexual network. Real-world sexual networks also depend on biased mixing based on age, location, and social and work activities. We describe an approach to use a broad range of social activity data to generate possible heterosexual networks. We start with a large-scale simulation of thousands of people in a city as they go through their daily activities, including work, school, shopping, and activities at home. We extract a social network from these activities where the nodes are the people, and the edges indicate a social interaction, such as working in the same location. This social network captures the correlations between people of different ages, living in different locations, their economic status, and other demographic factors. We use the social contact network to define a bipartite heterosexual network that is embedded within an extended social network. The resulting sexual network captures the biased mixing inherent in the social network, and models based on this pairing of networks can be used to investigate novel intervention strategies based on the social contacts among infected people. We illustrate the approach in a model for the spread of chlamydia in the heterosexual network representing the young sexually active community in New Orleans.

Using an agent-based sexual-network model to analyze the impact of mitigation efforts for controlling chlamydia

Chlamydia trachomatis (Ct) is the most reported sexually transmitted infection in the United States, with a major cause of infertility, pelvic inflammatory disease, and ectopic pregnancy among women. Despite decades of screening women for Ct, rates increase among young African Americans (AA). We create and analyze a heterosexual agent-based network model to help understand the spread of Ct. We calibrate the model parameters to agree with survey data showing Ct prevalence of 12% of the women and 10% of the men in the 15-25 year-old AA in New Orleans, Louisiana. Our model accounts for both long-term and casual partnerships. The network captures the assortative mixing of individuals by preserving the joint-degree distributions observed in the data. We compare the effectiveness of intervention strategies based on randomly screening men, notifying partners of infected people, which includes partner treatment, partner screening, and rescreening for infection. We compare the difference between treating partners of an infected person both with and without testing them. We observe that although increased Ct screening, rescreening, and treating most of the partners of infected people will reduce the prevalence, these mitigations alone are not sufficient to control the epidemic. The current practice is to treat the partners of an infected individual without first testing them for infection. The model predicts that if a sufficient number of the partners of all infected people are tested and treated, then there is a threshold condition where the epidemic can be mitigated. This threshold results from the expanded treatment network created by treating an individual's infected partners' partners. Although these conclusions can help design future Ct mitigation studies, we caution the reader that these conclusions are for the mathematical model, not the real world, and are contingent on the validity of the model assumptions.

Effect of screening young men for Chlamydia trachomatis on the rates among women: a network modelling study for high-prevalence communities

OBJECTIVE

Chlamydia trachomatis (Ct) is the most commonly reported sexually transmitted infection in the USA and causes important reproductive morbidity in women. The Centers for Disease Control and Prevention recommend routine screening of sexually active women under age 25 but not among men. Despite three decades of screening women, chlamydia prevalence in women remains high. Untested and untreated men can serve as a reservoir of infection in women, and male-screening based intervention can be an effective strategy to reduce infection in women. We assessed the impact of screening men on the Ct prevalence in women.

DESIGN

We created an individual-based network model to simulate a realistic chlamydia epidemic on sexual contact networks for a synthetic population (n=5000). The model is calibrated to the ongoing routine screening among African American (AA) women in the USA and detailed a male-screening programme, Check It, that bundles best practices for Ct control. We used sensitivity analysis to quantify the relative importance of each intervention component.

SETTING

Community-based venues in New Orleans, Louisiana, USA.

PARTICIPANTS

Heterosexual AA men, aged 15 to 24, who had sex with women in the past 2 months.

INTERVENTION

Venue-based screening, expedited index treatment, expedited partner treatment and rescreening.

RESULTS

We estimate that by annually screening 7.5% of the AA male population in the age-range, the chlamydia prevalence would be reduced relatively by 8.1% (95% CI 5.9% to 10.4%) in AA women and 8.8% (95% CI 6.9% to 10.8%) in AA men. Each man screened could prevent 0.062 (95% CI 0.030 to 0.094) cases in men and 0.204 (95% CI 0.143 to 0.267) cases in women. The model suggested the importance of intervention components ranked from high to low as venue-based screening, expedited index treatment, expedited partner treatment and rescreening.

CONCLUSION

The findings indicated that male-screening has the potential to substantially reduce the prevalence among women in high-prevalence communities.

Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases

The spread of an infectious disease is sensitive to the contact patterns in the population and to precautions people take to reduce the transmission of the disease. We investigate the impact that different mixing assumptions have on the spread an infectious disease in an age-structured ordinary differential equation model. We consider the impact of heterogeneity in susceptibility and infectivity within the population on the disease transmission. We apply the analysis to the spread of a smallpox-like disease, derive the formula for the reproduction number, ~0, and based on this threshold parameter, show the level of human behavioral change required to control the epidemic. We analyze how different mixing patterns can affect the disease prevalence, the cumulative number of new infections, and the final epidemic size. Our analysis indicates that the combination of residual immunity and behavioral changes during a smallpox-like disease outbreak can play a key role in halting infectious disease spread; and that realistic mixing patterns must be included in the epidemic model for the predictions to accurately reflect reality.

Modelling HIV/AIDS and monkeypox co-infection

During the last two decades, reports on emerging human monkeypox outbreaks in Africa and North America have reminded us that beside the eradicated smallpox there are other pox viruses that have great potential to harm people. A deterministic model for the co-infection of HIV/AIDS and monkeypox is formulated and analysed. The endemic equilibria are shown to be locally and globally asymptotically stable using the Centre Manifold theory and the Lyapunov function approach, respectively. Analysis of the basic reproduction numbers and numerical simulations suggest that an increase in the number of monkeypox in the animal species results in an increase of the number of people having monkeypox. Threshold conditions that determine the competitive outcomes of the two diseases are provided. Furthermore, numerical simulations using a set of reasonable parameter values support the claim that HIV infection greatly enhances monkeypox infection and vice versa.

Improving the damage accumulation in a biomechanical bone remodelling model

We extend, reformulate and analyse a phenomenological model for bone remodelling. The original macrobiomechanical model (MBM), proposed by Hazelwood et al. [J Biomech 2001; 34:299-308], couples a population equation for the cellular activities of the basic multicellular units (BMUs) in the bone and a rate equation to account for microdamage and repair. We propose to account for bone failure under severe overstressing by incorporating a Paris-like power-law damage accumulation term. The extended model agrees with the Hazelwood et al. predictions when the bone is under-stressed, and allows for suitably loaded bones to fail, in agreement with other MBM and experimental data regarding damage by fatigue. We numerically solve the extended model using a convergent algorithm and show that for unchanging loads, the stationary solution captures fully the model behaviour. We compute and analyse the stationary solutions. Our analysis helps guide additional extensions to this and other BMU activity based models.

Estimation of the reproduction number of dengue fever from spatial epidemic data

Dengue, a vector-borne disease, thrives in tropical and subtropical regions worldwide. A retrospective analysis of the 2002 dengue epidemic in Colima located on the Mexican central Pacific coast is carried out. We estimate the reproduction number from spatial epidemic data at the level of municipalities using two different methods: (1) Using a standard dengue epidemic model and assuming pure exponential initial epidemic growth and (2) Fitting a more realistic epidemic model to the initial phase of the dengue epidemic curve. Using Method I, we estimate an overall mean reproduction number of 3.09 (95% CI: 2.34,3.84) as well as local reproduction numbers whose values range from 1.24 (1.15,1.33) to 4.22 (2.90,5.54). Using Method II, the overall mean reproduction number is estimated to be 2.0 (1.75,2.23) and local reproduction numbers ranging from 0.49 (0.0,1.0) to 3.30 (1.63,4.97). Method I systematically overestimates the reproduction number relative to the refined Method II, and hence it would overestimate the intensity of interventions required for containment. Moreover, optimal intervention with defined resources demands different levels of locally tailored mitigation. Local epidemic peaks occur between the 24th and 35th week of the year, and correlate positively with the final local epidemic sizes (rho=0.92, P-value<0.001). Moreover, final local epidemic sizes are found to be linearly related to the local population size (P-value<0.001). This observation supports a roughly constant number of female mosquitoes per person across urban and rural regions.

Modelling the transmission dynamics of acute haemorrhagic conjunctivitis: application to the 2003 outbreak in Mexico

We model an outbreak of acute haemorrhagic conjunctivitis (AHC) using a simple epidemic model that includes susceptible, infectious, reported, and recovered classes. The model's framework considers the impact of underreporting and behaviour changes on the transmission rate and is applied to a recent epidemic of AHC in Mexico, using a fit to the cumulative number of cases to estimate model parameters, which agree with those derived from clinical studies. The model predicts a 'mean time from symptomatic onset to diagnosis' of 1.43 days (95 per cent CI: 1-2.5) and that the final size of the Mexican epidemic was underreported by 39 per cent. We estimate that a primary infectious case generates approximately 3 secondary cases (R0* = 2.64, SD 0.65). We explore the impact of interventions on the final epidemic size, and estimate a 36 per cent reduction in the transmission rate due to behaviour changes. The effectiveness of the behaviour changes in slowing the epidemic is evident at 21.90 (SD 0.19) days after the first reported case. Results therefore support current public health policy including expeditious announcement of the outbreak and public health information press releases that instruct individuals on avoiding contagion and encourage them to seek diagnosis in hospital clinics.

Human-mediated foot-and-mouth disease epidemic dispersal: disease and vector clusters

Disease clusters were retrospectively explored at national level using a geo-referenced dataset from the 2001 Uruguayan Foot-and-Mouth Disease (FMD) epidemic. Disease location and time (first 11 epidemic weeks) were analysed across 250 counties (of which 160 were infected), without and with control for human mobility related factors (human population and road densities). The null hypothesis of random disease distribution over space and/or time was assessed with: (i) purely temporal; (ii) purely spatial; and (iii) space/time tests. At least within epidemic weeks 2 and 6, a principal disease cluster was observed in 33 contiguous counties (P < 0.01). Two secondary clusters, located at >100 km from each other, were also observed (P < 0.01). The purely spatial test that controlled for human population density identified two non-contiguous clusters (P < 0.01). Space and time analysis also revealed the same 33 counties as members of the principal cluster, of which 31 were also clustered when human population was controlled (P < 0.01). No clusters were reported by the spatial test when road density was assessed. The hypothesis that human mobility related factors autocorrelate with disease was empirically supported by two pieces of information: (i) removal of human population/road densities eliminated >93.9% of the counties included in the principal disease cluster; and (ii) statistically significant correlations (P < 0.05) were observed in the first three epidemic weeks between road density and the number of cases. Clusters where human population density was associated with 47% greater number of cases/sq. km than that of the principal cluster indicated possible roles as disease vectors (vector clusters). Selective control policy in vector clusters is recommended. Periodic (i.e. weekly) cluster and correlation analyses of both disease and other covariates may facilitate disease surveillance and help design space-specific control policy.

Estimation of the reproductive number of the Spanish flu epidemic in Geneva, Switzerland

The 1918 influenza pandemic known as the "Spanish Flu" has been the worst in recent history with estimated worldwide mortality ranging from 20 to 100 million deaths. Using epidemic modeling and hospital notification data during the 1918 influenza pandemic in the Canton of Geneva, Switzerland, we estimated the reproductive numbers of the first and second waves of influenza infection to be R(1)=1.49 (95% CI: 1.45-1.53) and R(2)=3.75 (95% CI: 3.57-3.93), respectively. Our estimates indicate that containment of the next influenza pandemic could require strict interventions that include effective isolation strategies in hospitals and reductions in the susceptibility of the general population.

Transmission dynamics of the great influenza pandemic of 1918 in Geneva, Switzerland: Assessing the effects of hypothetical interventions

Recurrent outbreaks of the avian H5N1 influenza virus in Asia represent a constant global pandemic threat. We characterize and evaluate hypothetical public health measures during the 1918 influenza pandemic in the Canton of Geneva, Switzerland. The transmission rate, the recovery rate, the diagnostic rate, the relative infectiousness of asymptomatic cases, and the proportion of clinical cases are estimated through least-squares fitting of the model to epidemic curve data of the cumulative number of hospital notifications. The latent period and the case fatality proportion are taken from published literature. We determine the variance and identifiability of model parameters via a simulation study. Our epidemic model agrees well with the observed epidemic data. We estimate the basic reproductive number for the spring wave R1;=1.49 (95% CI: 1.45-1.53) and the reproductive number for the fall wave R2;=3.75 (95% CI: 3.57-3.93). In addition, we estimate the clinical reporting for these two waves to be 59.7% (95% CI: 55.7-63.7) and 83% (95% CI: 79-87). We surmise that the lower reporting in the first wave can be explained by a lack of initial awareness of the epidemic and the relative higher severity of the symptoms experienced during the fall wave. We found that effective isolation measures in hospital clinics at best would only ensure control with probability 0.87 while reducing the transmission rate by >76.5% guarantees stopping an epidemic.

Predicting scorpion sting incidence in an endemic region using climatological variables

Scorpionism is a public health problem in several regions of the world. The highest mortality, with over 1000 deaths per year, has been reported in Mexico. We analysed the significance of climatological variables to predict the incidence of scorpion stings in humans in the state of Colima (Mexico) for the years 2000-2001. The pluvial precipitation (mm), the evaporation (mm), and the mean, maximum, and minimum temperatures (degrees C) were obtained from local meteorological offices. There are approximately 3 stings/year per 1000 people in municipalities of Colima and Villa de Alvarez and about 18-30 stings/year per 1000 people in the rest of the municipalities. There is very little rain and there are few stings in the winter when the minimum temperature is below about 16 degrees C. The number of scorpion stings is independent of the actual rainfall when this is above 30 mm/month. Using multiple linear regression, we used a backward model selection procedure to estimate that the minimum temperature is correlated with scorpion sting incidence with a statistically significance of 95%. We briefly discuss the application of predictive models of scorpion sting incidence in the appropriate allocation of antivenom serum in hospital clinics.

The role of spatial mixing in the spread of foot-and-mouth disease

A model of epidemic dispersal (based on the assumption that susceptible cattle were homogeneously mixed over space, or non-spatial model) was compared to a partially spatially explicit and discrete model (the spatial model), which was composed of differential equations and used geo-coded data (Euclidean distances between county centroids). While the spatial model accounted for intra- and inter-county epidemic spread, the non-spatial model did not assess regional differences. A geo-coded dataset that resembled conditions favouring homogeneous mixing assumptions (based on the 2001 Uruguayan foot-and-mouth disease epidemic), was used for testing. Significant differences between models were observed in the average transmission rate between farms, both before and after a control policy (animal movement ban) was imposed. They also differed in terms of daily number of infected farms: the non-spatial model revealed a single epidemic peak (at, approximately, 25 epidemic days); while the spatial model revealed two epidemic peaks (at, approximately, 12 and 28 days, respectively). While the spatial model fitted well with the observed cumulative number of infected farms, the non-spatial model did not (P<0.01). In addition, the spatial model: (a) indicated an early intra-county reproductive number R of approximately 87 (falling to <1 within 25 days), and an inter-county R<1; (b) predicted that, if animal movement restrictions had begun 3 days before/after the estimated initiation of such policy, cases would have decreased/increased by 23 or 26%, respectively. Spatial factors (such as inter-farm distance and coverage of vaccination campaigns, absent in non-spatial models) may explain why partially explicit spatial models describe epidemic spread more accurately than non-spatial models even at early epidemic phases. Integration of geo-coded data into mathematical models is recommended.

Effects of behavioral changes in a smallpox attack model

The impact of individual and community behavioral changes in response to an outbreak of a disease with high mortality is often not appreciated. Response strategies to a smallpox bioterrorist attack have focused on interventions such as isolation of infectives, contact tracing, quarantine of contacts, ring vaccination, and mass vaccination. We formulate and analyze a mathematical model in which some individuals lower their daily contact activity rates once an epidemic has been identified in a community. Transmission parameters are estimated from data and an expression is derived for the effective reproduction number. We use computer simulations to analyze the effects of behavior change alone and in combination with other control measures. We demonstrate that the spread of the disease is highly sensitive to how rapidly people reduce their contact activity rates and to the precautions that the population takes to reduce the transmission of the disease. Even gradual and mild behavioral changes can have a dramatic impact in slowing an epidemic. When behavioral changes are combined with other interventions, the epidemic is shortened and the number of smallpox cases is reduced. We conclude that for simulations of a smallpox outbreak to be useful, they must consider the impact of behavioral changes. This is especially true if the model predictions are being used to guide public health policy.

The basic reproductive number of Ebola and the effects of public health measures: the cases of Congo and Uganda

Despite improved control measures, Ebola remains a serious public health risk in African regions where recurrent outbreaks have been observed since the initial epidemic in 1976. Using epidemic modeling and data from two well-documented Ebola outbreaks (Congo 1995 and Uganda 2000), we estimate the number of secondary cases generated by an index case in the absence of control interventions R0. Our estimate of R0 is 1.83 (SD 0.06) for Congo (1995) and 1.34 (SD 0.03) for Uganda (2000). We model the course of the outbreaks via an SEIR (susceptible-exposed-infectious-removed) epidemic model that includes a smooth transition in the transmission rate after control interventions are put in place. We perform an uncertainty analysis of the basic reproductive number R0 to quantify its sensitivity to other disease-related parameters. We also analyse the sensitivity of the final epidemic size to the time interventions begin and provide a distribution for the final epidemic size. The control measures implemented during these two outbreaks (including education and contact tracing followed by quarantine) reduce the final epidemic size by a factor of 2 relative the final size with a 2-week delay in their implementation.

Scaling laws for the movement of people between locations in a large city

Large scale simulations of the movements of people in a "virtual" city and their analyses are used to generate insights into understanding the dynamic processes that depend on the interactions between people. Models, based on these interactions, can be used in optimizing traffic flow, slowing the spread of infectious diseases, or predicting the change in cell phone usage in a disaster. We analyzed cumulative and aggregated data generated from the simulated movements of 1.6 x 10(6) individuals in a computer (pseudo-agent-based) model during a typical day in Portland, Oregon. This city is mapped into a graph with 181,206 nodes representing physical locations such as buildings. Connecting edges model individual's flow between nodes. Edge weights are constructed from the daily traffic of individuals moving between locations. The number of edges leaving a node (out-degree), the edge weights (out-traffic), and the edge weights per location (total out-traffic) are fitted well by power-law distributions. The power-law distributions also fit subgraphs based on work, school, and social/recreational activities. The resulting weighted graph is a "small world" and has scaling laws consistent with an underlying hierarchical structure. We also explore the time evolution of the largest connected component and the distribution of the component sizes. We observe a strong linear correlation between the out-degree and total out-traffic distributions and significant levels of clustering. We discuss how these network features can be used to characterize social networks and their relationship to dynamic processes.

Compacton solutions in a class of generalized fifth-order Korteweg-de Vries equations

Solitons play a fundamental role in the evolution of general initial data for quasilinear dispersive partial differential equations, such as the Korteweg-de Vries (KdV), nonlinear Schrödinger, and the Kadomtsev-Petviashvili equations. These integrable equations have linear dispersion and the solitons have infinite support. We have derived and investigate a new KdV-like Hamiltonian partial differential equation from a four-parameter Lagrangian where the nonlinear dispersion gives rise to solitons with compact support (compactons). The new equation does not seem to be integrable and only mass, momentum, and energy seem to be conserved; yet, the solitons display almost the same modal decompositions and structural stability observed in integrable partial differential equations. The compactons formed from arbitrary initial data, are nonlinearly self-stabilizing, and maintain their coherence after multiple collisions. The robustness of these compactons and the inapplicability of the inverse scattering tools, that worked so well for the KdV equation, make it clear that there is a fundamental mechanism underlying the processes beyond integrability. We have found explicit formulas for multiple classes of compact traveling wave solutions. When there are more than one compacton solution for a particular set of parameters, the wider compacton is the minimum of a reduced Hamiltonian and is the only one that is stable.

The origin of acquired immune deficiency syndrome: Darwinian or Lamarckian?

The subtypes of human immunodeficiency virus type 1 (HIV-1) group M exhibit a remarkable similarity in their between-subtype distances, which we refer to as high synchrony. The shape of the phylogenetic tree of these subtypes is referred to as a sunburst to distinguish it from a simple star phylogeny. Neither a sunburst pattern nor a comparable degree of symmetry is seen in a natural process such as in feline immunodeficiency virus evolution. We therefore have undertaken forward-process simulation studies employing coalescent theory to investigate whether such highly synchronized subtypes could be readily produced by natural Darwinian evolution. The forward model includes both classical (macro) and molecular (micro) epidemiological components. HIV-1 group M subtype synchrony is quantified using the standard deviation of the between-subtype distances and the average of the within-subtype distances. Highly synchronized subtypes and a sunburst phylogeny are not observed in our simulated data, leading to the conclusion that a quasi-Lamarckian, punctuated event occurred. The natural transfer theory for the origin of human acquired immune deficiency syndrome (AIDS) cannot easily be reconciled with these findings and it is as if a recent non-Darwinian process took place coincident with the rise of AIDS in Africa.

Combinations of clozapine and phencyclidine: effects on drug discrimination and behavioral inhibition in rats

Phencyclidine (PCP) produces psychotomimetic effects in humans that resemble schizophrenia symptoms. In an effort to screen compounds for antipsychotic activity, preclinical researchers have investigated whether these compounds block PCP-induced behaviors in animals. In the present study, the atypical antipsychotic clozapine was tested in combination with an active dose of PCP in two-lever drug discrimination and mixed signalled-unsignalled differential-reinforcement-of-low-rates (DRL) procedures. PCP produced distinctive effects in each task: it substituted for the training dose in PCP discrimination and it increased the number of responses with short (<3 s) interresponse times as well as increasing overall response rates in the DRL schedule. Acute dosing with clozapine failed to alter the behavioral effects of PCP in either procedure even when tested up to doses that produced pharmacological effects alone. These results suggest that acute dosing with clozapine would not affect behaviors most closely associated with PCP intoxication. Further, they bring into question the utility of using PCP combination procedures in animals to screen for antipsychotic potential. Since chronic dosing is required for therapeutic efficacy of antipsychotics, future studies should focus on investigation of chronic dosing effects of these drugs in combination with PCP.

The initialization and sensitivity of multigroup models for the transmission of HIV

We study two multigroup mathematical models of the spread of HIV. In the differential infectivity model, the infected population is divided into groups according to their infectiousness, and HIV is primarily spread by a small, highly infectious, group of superspreaders. In the staged-progression model, every infected individual goes through a series of infection stages and the virus is primarily spread by individuals in an initial highly infectious stage or in the late stages of the disease. We demonstrate the importance of choosing appropriate initial conditions, and define a new approach to distributing the initial population among the subgroups so as to minimize the artificial transients in the solutions due to unbalanced initial conditions. We demonstrate that the rate of removal in and out of a population is an important, yet often neglected, effect. We also illustrate the importance of distinguishing between the number of partners a person has and the number of contacts per partner. By assuming that people with many partners have fewer contacts per partner than people with few partners, we found that the epidemic is less sensitive to the partner acquisition rate than one might expect. However, because the probability of transmission of HIV per contact is low, the epidemic is very sensitive to the number of contacts per partner. Modeling this distinction is particularly important when estimating the impact of programs which encourage people to have fewer sexual partners.

An intuitive formulation for the reproductive number for the spread of diseases in heterogeneous populations

The thresholds for mathematical epidemiology models specify the critical conditions for an epidemic to grow or die out. The reproductive number can provide significant insight into the transmission dynamics of a disease and can guide strategies to control its spread. We define the mean number of contacts, the mean duration of infection, and the mean transmission probability appropriately for certain epidemiological models, and construct a simplified formulation of the reproductive number as the product of these quantities. When the spread of the epidemic depends strongly upon the heterogeneity of the populations, the epidemiological models must account for this heterogeneity, and the expressions for the reproductive number become correspondingly more complex. We formulate several models with different heterogeneous structures and demonstrate how to define the mean quantities for an explicit expression for the reproductive number. In complex heterogeneous models, it seems necessary to define the reproductive number for each structured subgroup or cohort and then use the average of these reproductive numbers weighted by their heterogeneity to estimate the reproductive number for the total population.

The differential infectivity and staged progression models for the transmission of HIV

Recent studies of HIV RNA in infected individuals show that viral levels vary widely between individuals and within the same individual over time. Individuals with higher viral loads during the chronic phase tend to develop AIDS more rapidly. If RNA levels are correlated with infectiousness, these variations explain puzzling results from HIV transmission studies and suggest that a small subset of infected people may be responsible for a disproportionate number of infections. We use two simple models to study the impact of variations in infectiousness. In the first model, we account for different levels of virus between individuals during the chronic phase of infection, and the increase in the average time from infection to AIDS that goes along with a decreased viral load. The second model follows the more standard hypothesis that infected individuals progress through a series of infection stages, with the infectiousness of a person depending upon his current disease stage. We derive and compare threshold conditions for the two models and find explicit formulas of their endemic equilibria. We show that formulas for both models can be put into a standard form, which allows for a clear interpretation. We define the relative impact of each group as the fraction of infections being caused by that group. We use these formulas and numerical simulations to examine the relative importance of different stages of infection and different chronic levels of virus to the spreading of the disease. The acute stage and the most infectious group both appear to have a disproportionate effect, especially on the early epidemic. Contact tracing to identify super-spreaders and alertness to the symptoms of acute HIV infection may both be needed to contain this epidemic.

Threshold conditions for the spread of the HIV infection in age-structured populations of homosexual men

The age-structure of a population, and the distribution of sexual behavior according to age, are significant factors determining the spread of the AIDS epidemic. The threshold conditions for age-structured models account for life-history information, and thus differ significantly from their age-independent counterparts. We examine the threshold conditions for four general age-structured models of the spread of HIV in a homosexual population: three with random partner selection and one with biased partner selection. We consider both discrete and continuous risk groups, and the duration of infection. Susceptibility and infectiousness are treated separately, and the infectivity varies with duration of infection. Through specific examples, we examine the sensitivity of the threshold conditions to the population age-structure and the shape of the infectivity profile. The effects of each are of the same order of magnitude.

Risk behavior-based model of the cubic growth of acquired immunodeficiency syndrome in the United States

The cumulative number of cases of acquired immunodeficiency syndrome (AIDS) in the United States has grown as the cube of time rather than exponentially. We explain this by interactions involving partner choice and sexual frequency in a risk-behavior model with biased mixing. This leads to a saturation wave of infection moving from high- to low-risk groups. If this description is correct, then the decreasing growth rate of AIDS cases is not due to behavior changes; rather it is due to the intrinsic epidemiology of the disease.

The need for national HIV databases

Researchers and public health officials involved in surveying and forecasting the course of the HIV epidemic require complete and unfiltered information from many sources. Governments should respond by establishing national HIV databases.

Nonlinear pattern selection in a mechanical model for morphogenesis

We present a numerical study of the nonlinear mechanical model for morphogenesis proposed by Oster et al. (1983) with the aim of establishing the pattern forming capability of the model. We present a technique for mode selection based on linear analysis and show that, in many cases, it is a reliable predictor for nonlinear mode selection. In order to determine the set of model parameters that can generate a particular pattern we develop a technique based on nonlinear least square fitting to a dispersion relation. As an application we present a scenario for sequential pattern formation of dermal aggregations in chick embryos which leads to the hexagonal array of cell aggregations observed in feather germ formation in vivo.